Higher structures in rational homotopy theory

TL;DR

This paper introduces higher algebraic structures in rational homotopy theory, connecting models like $C_ abla$- and $L_ abla$-algebras to classical theories, and explores their relation to Koszul duality and automorphisms of manifolds.

Contribution

It provides a concise overview of higher structures in rational homotopy theory, linking modern models to classical ones and establishing conditions for formality and coformality.

Findings

A space is formal iff its $L_ abla$-algebra is Koszul.

A space is coformal iff its $C_ abla$-algebra is Koszul.

Higher structures like Kontsevich's Lie graph complex are relevant in automorphisms of high-dimensional manifolds.

Abstract

These notes are based on a series of three lectures given (online) by the first named author at the workshop "Higher Structures and Operadic Calculus" at CRM Barcelona in June 2021. The aim is to give a concise introduction to rational homotopy theory through the lens of higher structures. The rational homotopy type of a simply connected space of finite type is modeled by a $C_\infty$-algebra structure on the rational cohomology groups, or alternatively an $L_\infty$-algebra structure on the rational homotopy groups. The first lecture is devoted to explaining these models and their relation to the classical models of Quillen and Sullivan. The second lecture discusses the relation between Koszul algebras, formality and coformality. The main result is that a space is formal if and only if the rational homotopy $L_\infty$-algebra is Koszul and, dually, a space is coformal if and only if the cohomology $C_\infty$-algebra is Koszul. For spaces that are both formal and coformal, this collapses to classical Koszul duality between Lie and commutative algebras. In the third lecture, we discuss certain higher structure in the rational homotopy theory of automorphisms of high dimensional manifolds, discovered by Berglund and Madsen. The higher structure in question is Kontsevich's Lie graph complex and variants of it.